# How many minutes in 1 day? There are 24*60 minutes in a day (ignor

How many minutes in 1 day?
There are 24*60 minutes in a day (ignoring the imperfections of the natural world, the Earth and Sun). So there are 24*60 valid 24 hour times (excluding seconds) on a digital clock.
Each of these can be rotated 4 ways, by 0,1,2 or 3 places. For example :
1 - 12:34
2 - 23:41
3 - 34:12
4 -4 1:23
In which case only 1 and 2 are valid 24 hour times.
How many of these 4*24*60 rotations are also valid minutes? (What is the fastest way to find this out?)
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Durst37

The easier part is the rotation by two digits: If the minutes are less than 24, this results in a time that is accurate, which happens in ${24}^{2}=576$ cases.
The ones of the minutes must be between 0 and 2, which is the case in 18 out of 60 minutes, because if we rotate by one to the right, they become the tens of the hours. The ones of the hours become the tens of the minutes, so they must be between 0 and 5, which is the case in 16 out of 24 hours. Thus the number of valid rotations is $18\cdot 16=288$.
There are 288 additional valid rotations since the number of valid rotations must equal the number of valid rotations if we rotate by one to the right or left.  Obviously if we don't rotate at all, all $24\cdot 60=1440$ times are valid, so the total is
slightly less than half.